Three results on transcendental meromorphic solutions of certain nonlinear differential equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf-Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers-Huxley, generalized equation of Camassa-Holm, generalized equation of Swift-Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

Solution of nonlinear partial differential equations from base equations

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that specific case of SEsM can be used in order to reproduce the methodology of the Inverse Scattering Transform Method for the case of the Burgers equation and Korteweg - de Vries equation. This specific case is connected to use of a specific case of Step. 2 of SEsM: representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter ε, solving the differential equations occurring from this representation by means of Fourier series and transition from the obtained solution for small values of ε to solution for arbitrary finite values of ε. Next, we discuss the application of composite functions in SEsM. We proof two propositions connected to obtaining solutions of nonlinear differential equations with polynomial nonlinearities by means of use of composite functions. We present several examples of applications of this methodology and obtain exact solutions of the generalized Korteweg - deVries equation, Olver equation, and several other equations. Next we discuss the most simple version of SEsM: the Modified Method of Simplest Equation (MMSE). We start with the role of the simplest equation and discuss the several cases of simplest equations such as nonlinear ordinary differential equations called Riccati equation and Bernoulli equation. The theory is illustrated by obtaining exact solution of various nonlinear partial differential equations such as Newel-Whitehead equation, FitzHugh-Nagumo equation, etc. MMSE is further illustrated by obtaining exact solutions of many equations such as Swift-Hohenberg equation, Rayleigh equation, Huxley equation. Special attention is given to the process of obtaining of balance equations in the MMSE. This process is illustrated by obtaining balance equations for several model nonlinear differential equations from the area of ecology and population dynamics. Among the discussed examples are the reaction-diffusion equation with density-dependent diffusion as well as the reaction-telegraph equation. Finally we obtain exact solution of two nonlinear model differential equations connected to the water wave propagation. These are the extended Korteweg-de Vries equation and the generalized Camassa-Holm equation. We close the discussion by several remark on the methodology and about the future plans connected to our research in this area.

Logistic function as solution of many nonlinear differential equations

when the values of y(x) are assigned at the m points a, a + 1, * * *, a + m -1. Obviously every equation (1) defines an infinite number of sequences: one for each set of values y(a), y(a + 1), * * *, y(a + m 1). We propose to obtain some of the properties of such sequences by studying the difference equations which define them. In the study of infinite sequences one is mainly interested in their ultimate behavior. Hence we are here interested in the behavior of the solutions of the difference equations, for integral values of x, in the neighborhood of infinity. Although, at present, it is not possible to write out explicitly the solutions of most non-linear difference equations, it seems that it might be feasible to determine whether a given equation defines sequences that approach zero as x becomes infinite by considering the solutions of the linear difference equation formed by omitting the non-linear terms. (We assume that at least one linear term appears.) For when y(x) approaches zero the linear terms are infinitesimals of the first order, while the non-linear terms are infinitesimals of higher order. Therefore, it would be expected that the behavior of such sequences is largely determined by the linear terms of the difference equation. With this idea in mind, we attempt to gain information about the sequences defined by a non-linear difference equation by considering the solution of the difference equation formed from its linear terms.3 It is clear that once we have criteria for determining when a difference equation defines sequences which approach zero we can easily determine whether one defines sequences that approach a constant limit a. For, if after the application of the transformation y(x) = g(x) + a, the transformed difference equa-

A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

Polygons of differential equations for finding exact solutions

Tests for determination of which nonlinear partial differential equations may have exact analytic nonlinear solutions of any of two types of hyperbolic functions or any of three types of Jacobian elliptic functions are presented. The Power Index Method is the principal method employed that extends the calculation of the power index for the most nonlinear terms to all terms in the nonlinear partial differential equations. An additional test is the identification of the net order of differentiation of each term in the nonlinear differential equations. The nonlinear differential equations considered are evolution equations. The tests extend the homogeneous balance condition that is necessary to conditions that may only be sufficient but are very simple to apply. Superposition of Jacobian elliptic functions is also presented with the introduction of a new basis that simplifies the calculations.

Seven common errors in finding exact solutions of nonlinear differential equations

Meromorphic solutions of nonlinear ordinary differential equations

Operators with regular singularities. One variable case - Operators with regular singularities. Several variables case - Formal and convergent solutions of singular partial differential equations - Local study of differential equations of the form xy' f(x,y) near x 0 - Holomorphic and singular solutions of non linear singular first order partial differential equations - Maillet's type theorems for non linear singular partial differential equations - Maillet's type theorems for non linear singular partial differential equations without linear part - Holomorphic and singular solutions of non linear singular partial differential equations - On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations - Maillet's type theorems for non linear singular integro-differential equations.

In Chap. 5, we explained how to apply the finite element method to nonlinear ordinary differential equations. We saw that calculating the finite element solution of nonlinear differential equations required us to solve a nonlinear system of algebraic equations and discussed how these algebraic equations could be solved. In this chapter, we explain how to apply the finite element method to nonlinear partial differential equations by combining: (i) the material on calculating the finite element solution of linear partial differential equations given in Chaps. 7– 10 and (ii) the material on calculating the finite element solution of nonlinear ordinary differential equations in Chap. 5.

Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method